Code
Optim
izationin
CD
MA
Systems
SennurU
lukus
AT
&T
Labs-R
esearch
Roy
D.Y
ates
WIN
LA
B,R
utgersU
niversity
ryates@w
inlab.rutgers.edu
1
Introduction�
Each
userhas
apow
er(p
i ),asignature
sequence(si )
anda
receiverfilter
(ci )
ps
k ,k
ps
i ,i
ps
j ,j
mo
bile
k
mo
bile
j
cccc
1ijN
mo
bile
i
......
BS
�
Whatare
thejointly
optimum
pi ,si ,c
i foralli?
2
From
Continuous
Signalsto
Vectors
+1
-1+1
+1
-1-1+1
-1+1
-1
i (t)
si (t)
c
Pro
cessing
gain
= L
L =
T
b
Tc
si=
T
T
c
b
t t
�
Chip
sampled
receivedsignal:
r �
∑Ni �
1�p
i ai si �
n
�
Receiver
filtering:y
i �
r �
ci
�
This
work
isgeneralizable
toany
comm
unicationsystem
where
transmit
waveform
sare
representablein
afinite
dimensionalvector
space.3
ConventionalC
DM
ASystem
�
Generate
si randomly
(nosignature
optimization)
�
Use
matched
filtersc
i �
si (nofilter
optimization)
�
Controltransm
itterpow
er(SIR
-basedpow
ercontrol)
pi� n�
1� �
γ �
i
γi� n� p
i� n�
�
Power
controlconvergesto
componentw
isesm
allestpower
vectorw
here
alluserssatisfy
theirSIR
requirements
�
Furtherim
provementcan
beachieved
ifthe
receiverfilters
aredesigned
tosuppress
interference
4
JointP
ower
andR
eceiverF
ilterO
ptimization
�
S.Ulukus
andR
.D.Y
ates,“Adaptive
Power
Controland
MM
SE
InterferenceSuppression,”
AC
MJ.on
Wireless
Netw
orks.
�
Fixedsi (no
signatureoptim
ization)
�
Choose
powers
(pi )
andlinear
receiverfilters
(ci )
jointlyoptim
ally
�
Iterativealgorithm
–For
fixedpow
ers,updatefilters
tom
inimize
theM
SE(equivalently
maxim
izethe
SIR)
–M
MSE
multiuser
detectors
–For
fixedreceiver
filtersupdate
powers
inthe
usualway
pi� n�
1� �γ �
i
γi� n� p
i� n�
�
Results
dependonly
onthe
signaturesequences;further
improvem
entcan
beachieved
ifthe
signaturesare
designedto
avoidinterference
5
Review
:Inform
ationT
heoreticC
DM
AC
apacity�
User
ihassignature
sequencesi ;letS �
s1���� sN .
�
Sumcapacity
ofa
generalmultiaccess
channel
Csum
�
max
� R1 ������
RN��� �
N∑i �
1 Ri
�
Csum
:m
aximum
totalnumber
ofbits
Nusers
cantransm
itonthe
uplink.
�
Forasingle
cellsynchronousC
DM
Asystem
,with
pi �
pforalli,(V
erdu)
Csum
�
12log� det� IL �
pσ2 SS �
��6
Maxim
izationof
theSum
Capacity
�
Form
atricesA
K �M
andB
M �
K
det� IK �
AB� �
det� IM �
BA�
�
CD
MA
sumcapacity
becomes
Csum
�
12log� det� IL �
pσ2 SS �
���
12log� det� IN �
pσ2 S �
S��
�
Tom
aximize
thesum
capacity(R
upf,Massey)
–If
N
�
L,S �
S �
IN(N
orthonormalsequences)
–If
N
�
L,SS �
�
NLIL
(NW
elchB
oundE
qualitysequences)
7
Review
:N
etwork
Capacity
�
Netw
orkcapacity:
Maxim
umnum
berof
admissible
usersgiven
processinggain
Land
SIRtargetβ
�
Nusers
areadm
issibleif
thereare
positivepow
ersp
i andsignature
sequencessi such
thatSIRi �
β
�
Netw
orkcapacity
with
MM
SEreceivers
(Visw
anath,Anantharam
,Tse)
N�
L1�
1β
�
The
maxim
umis
achievedw
ith
Equalreceived
powers:
pi �
pfor
alli
WB
Esignature
sequences:SS �
�� N� L� IL
8
Netw
orkC
apacityII
�
MM
SEreceiver
forthe
ithuser
is
ci �
�
pi B
1si
�
When
pi �
pfor
alli,B
�
pSS ��
σ2IL
�
With
WB
Esequences
SS ��
NLIL
ci �
αi si
scaledm
atchedfilters!
�
Netw
orkcapacity
with
matched
filters(V
iswanath,A
nantharam,Tse)
N
�
L1�
1β
�
Max
isachieved
with
equalrec’dpow
ersand
WB
Esequences
�
“Power
/signaturesequence
/receiverfilter
optimization”
problem
actuallyhas
two
degreesoffreedom
:pow
ersand
signatures.
9
Simple
example,L
!
2,N
!
1
10
Simple
example,L
!
2,N
!
2
11
Simple
example,L
!
2,N
!
3
12
Welch’s
Bound
�
Originally
alow
erbound
form
axi�
j� s �is
j� 2kin
asetof
unitenergyvectors
�
The
derivationuses
thelow
erbound:
N∑i �1
N∑j �
1 � s �
is
j� 2k�
N2
" L#
k
1k
$
�
Fork �
1,alow
erbound
forTotalSquared
Correlation:
TSC
�N∑i �
1
N∑j �1 � s �
is
j� 2�
N2
L
�
IfN
�
L,thebound
isloose.
ForN
orthonormalvectors,T
SC
�
N
�
IfN
�
L,thebound
isachieved
iffSS �
�
NLIL
(Massey,M
ittelholzer)
13
WB
ESequences,M
inimum
TSC
,andO
ptimality
�
Minim
umT
SCsequences:
–O
rthonormalsequences
forN
�
L
–W
BE
sequencesfor
N
�L
�
Fora
singlecellC
DM
Asystem
,minim
umT
SCsequences
maxim
ize
–Inform
ationtheoretic
sumcapacity
–N
etwork
capacity
�
Goal:A
simple
algorithmw
hichconverges
toa
setofm
inimum
TSC
sequences.
14
WB
ESequences
andM
inimum
TSC
�
Optim
alsignaturesare
minim
umT
SCsignatures
�
Startingpointfor
TSC
reduction:
TSC
�� s �
ksk� 2
%&'(
1
�2s �
k∑j) �
k sj s �
jsk �
∑i) �
k ∑j) �
k � s �
is
j� 2
�
Many
ways
toreduce
TSC
:
–e.g.
choosesk
tobe
eigenvectorof∑
j) �k s
j s �j
with
min
eigenvalue
15
An
IterativeT
SCR
eductionA
lgorithm�
Method:R
eplacesk
with
ck �
A
1k
sk
� s �kA
2k
sk� 1* 2
where
Ak �
∑j) �
k sj s �
j �a
2IL .
�
ck
isa
generalizednorm
alizedM
MSE
filterfor
userk.
�
PracticalImplem
entation:
–U
sea
blindadaptive
MM
SEdetector
foreach
user.
–W
henreceiver
filterfor
userk
converges,transmitfilter
coefficients
backto
thetransm
itter
16
Algorithm
Properties
�
Algorithm
:Replace
skw
ithM
MSE
filterck
Old
Signatures:S �
s1���� sk
1 sk sk#
1���� sN
New
Signatures:S +
� s1 ��� sk
1 ck sk#
1���� sN
�
Theorem
:TSC� S +
� �
TSC� S�
andT
SC� S +� �
TSC� S�
iffc
k �
sk .
17
The
IterativeA
lgorithm
sN (n-1)
s s(n-1)
1(n-1)2
sN (n-1)
s s1(n-1)2
sN (n-1)
s s(n)
12
sN (n-1)
s s(n)
1(n)2
sN (n)
s s(n)
1(n)2
sN (n)
s s1(n)2
(n)
(n)
(n+1)
iteration n
step 1step 2
step Nstep (N
-1)
iteration (n+1)
SS
SS
SS
(n-1)(n)
1(n)
2(n)
N-1
N (n)(n+1)
1
(n)S =
iteration (n-1)end of. . .
. . .
. . .
. . .
. . .
. . .
. . .
�
TSC� n ,
1� �
TSC
1� n� �����
TSC
N
1� n� �
TSC
N� n� �
TSC� n�
18
Algorithm
Convergence
�
TSC� n�
isdecreasing
andlow
erbounded �-
TSC� n�
converges
�
TSC� n�
converges �-
S� n�/.S
�
Does
TSC
reachglobalm
inimum
?19
Properties
ofthe
Fixed
Point
�
LetS �
limn0
∞S� n�
�
N
�
L:S
convergesto
S �
S �
IN
–Sufficientcondition: initialsignature
sequencesS� 0�
arelinearly
independent
�
N
�
L:S
convergesto
SS ��
NLIL
–Sufficientcondition: L
ofN
initialsignaturesequences
S� 0�
are
linearlyindependent
–N
ecessarycondition:S� 0�
doesnothave
orthogonalsubsets
�
Forboth
casessufficientconditions
canbe
relaxed
20
Min/M
axE
igenvaluesofS 1
2 n3 S2 n3
andT
SC
2 n3
02
46
810
0
0.5 1
1.5 2
2.5 3
iteration (n)
min. and max. eigenvalues of STS
min. and m
ax. e.v. of S TS for N=5m
in. and max. e.v. of S TS for N=10
1
02
46
810
4 6 8 10 12 14 16 18 20
iteration (n)
TSC = trace(SSTSST)N=5
N=10
TSCN
N
�
Lcase:N
�
5 10and
L �
10
21
Min/M
axE
igenvaluesofS2 n3 S 1
2 n3
andT
SC
2 n3
02
46
810
0 1 2 3 4 5 6 7 8 9 10
iteration (n)
min. and max. eigenvalues of SST
N=20
N=30
N=40
N=50
min. and m
ax. e.v. of SS T
N/L
02
46
810
0 50
100
150
200
250
300
350
iteration (n)
TSC = trace(SSTSST)
N=20
N=30
N=40
N=50
TSCN
2/L
N
�
Lcase:N
�
20 30 40 50and
L �10
22
Conclusions
�
This
work
developeda
distributedalgorithm
foradapting
users’
signaturesthatcan
beused
forinterference
avoidance.
�
The
algorithmis
shown
toconverge
toa
setofusers’
signaturesthatare
optimalboth
interm
sof
information
theoreticcapacity
andnetw
ork
capacity.
�
The
algorithmcan
beim
plemented
usingfeedback
tothe
transmitter
from
anadaptive
MM
SEreceiver.
23
Work
inP
rogress
�
Extensions
toasynchronous
systems.
�
Analysis
ofm
ultipathchannels
�
Multiple
receivers(m
ulticellsystems)
�
Implem
entationbased
onblind
adaptivedetectors.
�
Effectiveness
inunlicensed
environments.
24